Anomalous ANITA air shower events and tau decays

Two unusual neutrino events in the Antarctic Impulse Transient Antenna (ANITA) appear to have been generated by air showers from a particle emerging from the Earth at angle similar to 25 degrees 35 degrees above the horizon. We evaluate the effective aperture for ANITA with a simplified detection model to illustrate the features of the angular dependence of expected events for incident standard model tau neutrinos and for sterile neutrinos that mix with tau neutrinos. We apply our sterile neutrino aperture results to a dark matter scenario with long-lived supermassive dark matter that decay to sterile neutrinolike particles. We find that for upgoing air showers from tau decays, from isotropic fluxes of standard model, sterile neutrinos, or other particles that couple to the tau through suppressed weak interaction cross sections cannot be responsible for the unusual events.